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A new exact representation of the error function of real arguments justifies an accurate and simple analytical approximation. Two of the most widely used functions in physical sciences are the error function erf(x) and its related complimentary error function erfc(x). These functions occur extensively in problems relating to diffusion, heat conduction, and probability. When the argument x is real, rational approximations for these functions provide a high accuracy. In addition, the Faddeeva function, which is a variant of the error function for the complex argument z = x + iy, is used extensively in nuclear physics and spectroscopy.